839.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 839

References:

erdosproblems.com/839
[Er78f] Erdős, P., Problems in number theory and combinatorics, Proc. Sixth Manitoba Conf. on Numerical Math. (1978), 35-58.
[Er92c] Erdős, P., Some of my favourite unsolved problems, J. Combin. Theory Ser. A (1992).

Imports (1)

FormalConjecturesUtil

def Erdos839.SumOfConsecutiveFree A sequence $a : \mathbb{N} \to \mathbb{N}$ is "sum-of-consecutive-free" if no term equals the sum of a contiguous block of earlier terms line 43 source
Source documentation
A sequence $a : \mathbb{N} \to \mathbb{N}$ is "sum-of-consecutive-free" if no term equals the sum of a contiguous block of earlier terms. That is, for all $i$, $a_i \neq a_j + a_{j+1} + \cdots + a_k$ for any $j \leq k < i$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: LE.le, LT.lt, Ne, Finset.sum, Finset.Icc
5 instances, coercions and auxiliary constants left out.
```
def SumOfConsecutiveFree (a : ℕ → ℕ) : Prop :=
…
```
theorem Erdos839.erdos_839.parts.i Documented declaration: i sorry line 55 source
Source documentation
Erdős Problem 839 (Part 1) [Er78f][Er92c]:
Let $1 \leq a_1 < a_2 < \cdots$ be a strictly increasing sequence of positive integers such that no $a_i$ is the sum of consecutive $a_j$ for $j < i$. Is it true that $\limsup a_n / n = \infty$?

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, StrictMono, Erdos839.SumOfConsecutiveFree, Eq, ENNReal, Filter.limsup, HDiv.hDiv, Filter.atTop, Top.top
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_839.parts.i : answer(sorry) ↔
    ∀ (a : ℕ → ℕ), (∀ n, 1 ≤ a n) → StrictMono a → SumOfConsecutiveFree a →
    atTop.limsup (fun n : ℕ => (a n : ℝ≥0∞) / n) = ⊤ := by
…
```
Mentions: Erdos839.SumOfConsecutiveFree
theorem Erdos839.erdos_839.parts.ii Documented declaration: ii sorry line 71 source
Source documentation
Erdős Problem 839 (Part 2, stronger) [Er78f][Er92c]:
Let $1 \leq a_1 < a_2 < \cdots$ be a strictly increasing sequence of positive integers such that no $a_i$ is the sum of consecutive $a_j$ for $j < i$. Is it true that $\lim_{x \to \infty} \frac{1}{\log x} \sum_{a_n < x} \frac{1}{a_n} = 0$?
This is equivalent to asking whether the range $\{a_1,a_2,\ldots\}$ has logarithmic density zero (see Set.HasLogDensity).

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, StrictMono, Erdos839.SumOfConsecutiveFree, Set.HasLogDensity, Set.range, Real
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_839.parts.ii : answer(sorry) ↔
    ∀ (a : ℕ → ℕ), (∀ n, 1 ≤ a n) → StrictMono a → SumOfConsecutiveFree a →
    Set.HasLogDensity (Set.range a) 0 := by
…
```
Mentions: Erdos839.SumOfConsecutiveFree, Set.HasLogDensity